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(a) The magnitude of the angular momentum about the origin of a particle of mass m moving with velocity v on a path that is a perpendicular distance d from the origin is given by m|v|d. Show that if r is the position of the particle then the vector J = r × mv represents the angular momentum. (b) Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection being represented by w. (i) Show that the velocity of the ith particle is Vi = @ X ri and that the total angular momentum J is J = Σm;[r}w − (r; · w)r;]. i

(a) The magnitude of the angular momentum about the origin of a particle of mass m moving with velocity v on a path that is a perpendicular distance d from the origin is given by m|v|d. Show that if r is the position of the particle then the vector J = r × mv represents the angular momentum. (b) Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection being represented by w. (i) Show that the velocity of the ith particle is Vi = @ X ri and that the total angular momentum J is J = Σm;[r}w − (r; · w)r;]. i

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(a) The magnitude of the angular momentum about the origin of a particle of mass m moving with velocity v on a path that is a perpendicular distance d from the origin is given by m/v|d. Show that if r is the position of the particle then the vector J =r × mv represents the angular momentum. (b) Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection being represented by w. (i) Show that the velocity of the ith particle is Vi = w X ri and that the total angular momentum J is J = Σm₁ [r}w - (r; · w)r;]. (ii) Show further that the component of J along the axis of rotation can be written as Iw, where I, the moment of inertia of the collection about the axis or rotation, is given by 1 = Σm₁p². Interpret pi geometrically. (iii) Prove that the total kinetic energy of the particles is 1².
Transcribed Image Text:(a) The magnitude of the angular momentum about the origin of a particle of mass m moving with velocity v on a path that is a perpendicular distance d from the origin is given by m/v|d. Show that if r is the position of the particle then the vector J =r × mv represents the angular momentum. (b) Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection being represented by w. (i) Show that the velocity of the ith particle is Vi = w X ri and that the total angular momentum J is J = Σm₁ [r}w - (r; · w)r;]. (ii) Show further that the component of J along the axis of rotation can be written as Iw, where I, the moment of inertia of the collection about the axis or rotation, is given by 1 = Σm₁p². Interpret pi geometrically. (iii) Prove that the total kinetic energy of the particles is 1².
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